Finite Group Databasebeta

← #186625

Subgroups — #186625

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#20734C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7C2×#1479C211yesmaximal, characteristic
8#8448C218yesmaximal, complemented
9C2×#557—24no—
10C2×#557—22no—
11C2×#557—24no—
12C2×#557—22no—
13C2×#557—24no—
14C2×#557—22no—
15C2×#557C418yescomplemented
16C2×#557C2214yes—
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125—21no—
20C22×#125—22no—
21C22×C4×Q8—21no—
22C22×C4×Q8—21no—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419—21no—
29#2285C2211yescharacteristic
30#1990C2214yes—
31#1988—22no—
32#1824—21no—
33#1823—22no—
34#1805—22no—
35#1803C2214yes—
36#1803C2214yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1546—24no—
41C2×#332C2214yes—
42#1487—24no—
43#1479—24no—
44C2×Q8⋊C8C2214yes—
45C2×Q8⋊C8C2214yes—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C23×Q8D411yescharacteristic
49C22×M16—21no—
50#557—216no—
51#557—216no—
52#557—216no—
53#557—216no—
54#557—28no—
55#557—216no—
56#557—216no—
57#557—28no—
58#557—28no—
59#557C2×C4116yes—
60C2×C4⋊Q8—28no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—28no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8C2×C414yes—
69C2×C4⋊Q8D414yes—
70C2×C4⋊Q8D412yes—
71C2×C4⋊Q8C2312yes—
72C2×#125—28no—
73C2×#125—216no—
74C2×#125—28no—
75C2×#125—24no—
76C2×#125—216no—
77C2×#125—28no—
78C2×#125—28no—
79C2×#125—24no—
80C2×C4×Q8—24no—
81C2×C4×Q8—28no—
82C2×C4×Q8—24no—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—24no—
86C2×C4×Q8—28no—
87C2×C4×Q8—24no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—21no—
91C2×C4×Q8—22no—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8C2×C414yes—
97C2×C4×Q8C2×C414yes—
98C2×C4×Q8D414yes—
99C2×C4×Q8D414yes—
100C2×C4×Q8D412yes—
101C2×C4×Q8D412yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×#117—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D412yes—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C42—22no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#431—22no—
119#431—22no—
120#431—22no—
121#431—22no—
122#423—22no—
123#423—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128#419—44no—
129C2×#103—24no—
130C2×#103—24no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8C2312yes—
135#384C2312yes—
136#382—21no—
137#380—21no—
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141#340—24no—
142#340—24no—
143#340—28no—
144#332—28no—
145Q8⋊C8—28no—
146Q8⋊C8—28no—
147C22×Q8—28no—
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—24no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8C22⋊C414yes—
165C22×Q8C22⋊C414yes—
166C22×Q8C2×D412yes—
167C22×Q8C2×D412yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—42no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C4⋊Q8—216no—
183C4⋊Q8—232no—
184C4⋊Q8—216no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8C22⋊C418yes—
193C4⋊Q8C22×C414yes—
194C4⋊Q8C2×D414yes—
195#125—232no—
196#125—264no—
197#125—216no—
198#125—232no—
199#125—232no—
200#125—216no—
201C4×Q8—216no—
202C4×Q8—216no—
203C4×Q8—232no—
204C4×Q8—232no—
205C4×Q8—216no—
206C4×Q8—216no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8C22⋊C418yes—
222C4×Q8C22⋊C418yes—
223C4×Q8C22×C414yes—
224C4×Q8C22×C414yes—
225C4×Q8C2×D414yes—
226C4×Q8C2×D414yes—
227#117—42no—
228#117—42no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22⋊C414yes—
258C2×C4⋊C4C22⋊C418yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—41no—
263C2×C42—28no—
264C2×C42—216no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—41no—
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2411yescharacteristic
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288#103—48no—
289#103—48no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—42no—
295#95—41no—
296#95—41no—
297C24#12011yescharacteristic
298C2×Q8—216no—
299C2×Q8—28no—
300C2×Q8—216no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—42no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—48no—
325C2×Q8—42no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—48no—
329C2×Q8C2×C22⋊C414yes—
330C2×Q8C2×C22⋊C414yes—
331C2×Q8#13718yes—
332C22×C4—28no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4#12012yes—
356C22×C4#12011yescharacteristic
357C22×C4#12011yescharacteristic
358C22×C4#12012yes—
359C22×C4#12012yes—
360C22×C4#12011yescharacteristic
361C22×C4#13512yes—
362C22×C4#13512yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368M16—84no—
369C2×C8—44no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—44no—
376C4⋊C4—232no—
377C4⋊C4—232no—
378C4⋊C4—232no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—28no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4C2×C22⋊C414yes—
399C4⋊C4C2×C22⋊C414yes—
400C4⋊C4C2×C22⋊C418yes—
401C22⋊C4—82no—
402C42—232no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—28no—
408C42—28no—
409C42—82no—
410C42C2×C22⋊C412yes—
411C42C2×C22⋊C412yes—
412C42C2×C22⋊C412yes—
413C42#12012yes—
414C42#12012yes—
415C42#12012yes—
416C42C23×C411yescharacteristic
417C42C22×D411yescharacteristic
418C42C22×D411yescharacteristic
419C42C22×D411yescharacteristic
420C23—22no—
421C23—22no—
422C23#37912yes—
423C23#44912yes—
424C23#45212yes—
425C23C2×#12011yescharacteristic
426Q8—216no—
427Q8—24no—
428Q8—28no—
429Q8—24no—
430Q8—24no—
431Q8—44no—
432Q8—44no—
433Q8—42no—
434Q8—44no—
435Q8—42no—
436Q8—44no—
437Q8—48no—
438Q8—416no—
439Q8#41918yescomplemented
440C2×C4—216no—
441C2×C4—28no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—48no—
463C2×C4—416no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4#37912yes—
468C2×C4#37912yes—
469C2×C4#37912yes—
470C2×C4#41614yes—
471C2×C4#41714yes—
472C2×C4#44914yes—
473C2×C4#44912yes—
474C2×C4#45214yes—
475C2×C4#45212yes—
476C2×C4C22×C22⋊C411yescharacteristic
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C22×C22⋊C411yescharacteristic
479C2×C4C2×#12011yescharacteristic
480C2×C4C2×#12011yescharacteristic
481C2×C4C2×#12011yescharacteristic
482C2×C4C2×#13511yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
485C2×C4C2×#13714yes—
486C2×C4C2×#13711yescharacteristic
487C8—82no—
488C8—82no—
489C8—82no—
490C22—24no—
491C22—24no—
492C22—22no—
493C22—22no—
494C22#178814yes—
495C22C2×#37911yescharacteristic
496C22C2×#44911yescharacteristic
497C22C2×#45211yescharacteristic
498C22#300412yes—
499C22#302012yes—
500C4—28no—
501C4—22no—
502C4—21no—
503C4—21no—
504C4—22no—
505C4—44no—
506C4—48no—
507C4—44no—
508C4—44no—
509C4—162no—
510C4#178814yes—
511C4C2×#41611yescharacteristic
512C4C2×#41912yes—
513C4C2×#41711yescharacteristic
514C4C2×#44911yescharacteristic
515C4C2×#45211yescharacteristic
516C4#300412yes—
517C2—24no—
518C2#1876012yes—
519C2#1876112yes—
520C2C2×#178811yescharacteristic
521C2#2059911yescharacteristic
522C2#2061711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #8448C2semidirect
15. C2×#557C4semidirect
439. Q8#419semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#5572
#218332
#207342
#206472
C2×#15462
C2×#14872
C2×#14792
#84482